>David C. Ullrich wrote:>> On Mon, 10 Dec 2012 21:05:31 +0000, José Carlos Santos>> <jcsantos@fc.up.pt> wrote:>> >>> Hi all,>>>>>> One of my students asked me today a question that I was unable to >>> answer. Let _f_ be an analytical function from (0,+oo) into [1,+oo) and >>> suppose that the integral of _f_ from 1 to +oo converges. Does it follow >>> that the series sum_n f(n) converges?>> >> Certainly not.>> >>> I don't think so, but I was unable >>> to find a counter-example. Any ideas?>> >> sum_n (1 + (x-n)^2)^{k(n)}>> >> gives a counterexample if k(n) -> infinity>> fast enough.>>Missing a negative sign before the k(n) ?

Yes, sorry.

>>Similar would be a sum of bell curves,>> sum_n exp(-(x-n)^2 k(n)) ,>>where k(n) is, say, n^4 .>>> Details in a few days after finals are done, if you remind me...>> >>> Best regards,>>>>>> Jose Carlos Santos